There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize:
Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. Then one can show (e.g. Huybrechts, complex geometry Prop. 4.3.8) that the curvature $F = \nabla^2 \in \mathcal{A}^{1,1}(M,\mathbb{C})$ is a (1,1)-Form with the property that

(i) $h(F_{X,Y}\sigma,\tau)=-h(\sigma,F_{X,Y}\tau) $.

Now, writing $F$ in local coordinates $(z_i)$ of $M$, we see $F=\sum_{i,j}F_{Z_i,\bar Z_j}dz_i \wedge d\bar z_j$ where

$ a_{ij}:=F_{Z_i,\bar Z_j}$

is a hermitian symmetric matrix.
Now my question is:

From the first equality (i) we deduce, that $ F_{X,Y} $ is a purely imaginary complex number. Why isn't this a contradiction to $ a_{ij} $ being hermitian symmetric? The matrix entries of $ a_{ij} $ are special $ F_{X,Y} $, so we get a matrix with purely imaginary entries which cannot be positive definite.